Before Fiji was ceded to Great Britain in 1874 there was quite a bit of documentation flowing backwards and forwards between various members of the British Government, and much of it was printed and distributed for the benefit of the Members of the House of Commons. Once Fiji became a colony there were various reports and correspondence tabled. Although rarely mentioning individuals by name it is very useful to historians.
The Parliamentary Papers for the British House of Commons have been digitised and categorised for the use of researchers. The website is http://parlipapers.chadwyck.co.uk but you need to have a login and password to enter it.
Fortunately, if you have a Library Card from the National Library of Australia you can access the site for free. Just go to the Library’s homepage and click on eResources in the top right hand corner. Here you can enter your Library Card number and your family name. If you don’t have a Library Card you can request one, and it will be posted within a couple of weeks.
Once you’ve logged in using your Library Card go down to Find a resource and type in ‘House of Commons’. Accept the terms and conditions. If you then Browse Subject Catalogue you need to get down to The dominions and colonies:
As you can see there are documents for other Pacific Islands as well. Here is the list of documents for the years 1801-1900 under the heading Fiji Islands:
1862  Fiji islands. Correspondence relative to the Fiji islands.
1871 (435) Fiji Islands. Return to an address of the honourable the House of Commons, dated 24 April 1871;–for, copies or extracts of correspondence and documents relating to the Feejee Islands, in so far as the same relate to their annexation to the colonial empire of this country, or otherwise affording protection to British subjects resident in those islands.”
1872 [C.509] Fiji Islands. Further correspondence relating to the Fiji Islands.
1873 (76) Fiji Islands (instructions to naval officer). Copy of any instructions that may have been sent to the naval officer commanding in the pacific relative to the line of conduct to be adopted by commanders of Her Majesty’s vessels towards the so-called government of the Fiji.
1873 (124) Fiji islands (correspondence with New South Wales). Copies of despatches to the Governor of New South Wales (subsequent to letter 88 of 3rd November 1371, published in parliamentary paper, no. 509, of 1872), respecting the acknowledgment of the government set up by a section of the white settlers in the Fiji islands, as well as of the minutes or correspondence which have passed between the Governor of New South Wales and his executive council on the same subject.
1873 (337) Fiji Islands. Copy of a despatch from Captain Chapman, of Her Majesty’s ship “dido,” to Commodore Stirling, 29th March 1873, with its enclosures, relative to the dispute between the Fiji Government and the white settlers of the district of Ba, in the Fiji Islands.
1874 [C.983] Fiji Islands. Copy of a letter addressed to Commodore Goodenough, R.N., and E. L. Layard, Esq., Her Majesty’s consul in Fiji, instructing them to report upon various questions connected with the Fiji Islands: with enclosures.
1874 [C.1011] Report of Commodore Goodenough and and Mr. Consul Layard on the offer of the cession of the Fiji Islands to the British crown.
1875 [C.1114] [C.1337] Correspondence respecting the colony of Fiji.
1876 (399) Fiji (measles). Copy of letter from the Admiralty to Commodore Hoskins, conveying their views on the alleged introduction of measles into Fiji by the officers of Her Majesty’s ship “Dido.”
1876 (408) Fiji (measles). Copy of the letter from the secretary of state for the colonies to the Governor of Fiji, communicating the views of Her Majesty’s Government as to the responsibility of the administrator of the colony and the acting colonial secretary for the introduction of measles into Fiji.
1876 [C.1404] [C.1624] Further correspondence respecting the colony of Fiji.
1877 [C.1826] Further correspondence relative to the colony of Fiji.
1877 [C.1880] Fiji. Correspondence in connexion with the native produce taxes in Fiji.
1878 (111) Polynesian labourers. Copies of ordinances introduced by Sir Arthur Gordon to regulate treatment of Polynesian labourers, and the introduction of Indian coolies into Fiji; and, of correspondence relating to these ordinances between the Colonial Office and the Government of Fiji.
1878 (285) Marriages (Fiji). A bill to remove doubts as to the validity of certain marriages solemnized in the islands of Fiji prior to their erection into a British colony.
1880 (411) Fiji (ship “Leonidas”). Copy or extracts of the correspondence which took place between Mr. Des Vœux, administrator of Fiji, and the Secretary of State for the Colonies, relative to the detention of the coolie ship “Leonidas” at Nasova in May 1879, in consequence of an outbreak of smallpox on board, and also any reports showing the successful efforts of the administrator to prevent the introduction of the disease into Fiji.
1883 [C.3584] [C.3815] Fiji. Correspondence relative to land claims in Fiji. Maps will be found at page 64.
1884-85 [C.4433] Fiji. Further correspondence respecting claims of German subjects to land in Fiji. (In continuation of [C.-3584] of April 1883 and [C.-3815] of August 1883.) A map will be found at page 78.
1884-85 [C.4434] Fiji. Correspondence relating to the native population of Fiji. Part I.–Native labour ordinances. Part II.–Condition of the native population.
1887 [C.5039] Fiji. Correspondence relating to the native population of Fiji. (In continuation of [C.–4434] May 1885.) Maps will be found at pages 40, 72, and 74.
1888 [C.5249-37] Her Majesty’s colonial possessions. No. 34. Fiji. Report on the blue book for 1887.
1890 [C.5897-9] Her Majesty’s colonial possessions. No. 79. Fiji. Report on the blue book for 1888. (In continuation of colonial possessions report no. 34.)
1890-91 [C.6221-5] Her Majesty’s colonial possessions. No. 116. Fiji. Report on the blue book for 1889. (In continuation of Colonial possessions report no. 79.)
1892 [C.6829] Colonial reports.–Annual. No. 45. Fiji. Annual report for 1890. (For report for 1889, see Colonial Report no. 116, Old Series.)
1893-94 [C.6857-22] Colonial reports.–Annual. No. 72. Fiji. Annual report for 1891. (For report for 1890, see Colonial Report [Annual] No. 45.)
1893-94 [C.6857-47] Colonial reports.–Annual. No. 97. Fiji. Annual report for 1892. (For report for 1891, see Colonial Report [Annual] No. 72.)
1895 [C.7629-10] Colonial reports.–Annual. No. 127. Fiji. Annual report for 1893. (For report for 1892, see colonial report [Annual] no. 97.)
1895 [C.7679] Fiji. Further correspondence respecting the affairs of Fiji and the native population. (In continuation of [C. 5039, April 1887].)
1896 [C.7944-5] Colonial reports.–Annual. No. 153. Fiji. Annual report for 1894. (For report for 1893, see colonial report [Annual] no. 127.)
1898 [C.8650-1] Colonial reports.–Annual. No. 203. Fiji. Report on trade for 1896.
1898 [C.8650-15] Colonial reports–annual. No. 217. Fiji. Annual report for 1896.
1899 [C.9046-12] Colonial reports–annual. No. 244. Fiji. Annual report for 1897. (For report for 1896, see no. 217.)
1899 [C.9498-2] Colonial reports–annual. No. 268. Fiji. Annual report for 1898. (For report for 1897, see no. 244)
1900 [Cd.354-2] Colonial reports–Annual. No. 296. Fiji. Report for 1899. (For report for 1898, see no. 268.)
The documents are all downloadable as PDF files, and some of them are quite large. Here is an example from the 1871 collection of documents and despatches related to the Fiji Islands “in so far as the same relate to their annexation to the colonial empire of this country, or otherwise affording protection to British subjects resident in those islands.”
These documents are indispensable to historians and are easily obtainable for Australian residents. Libraries and universities in other countries may have similar arrangements, so it’s worth checking.