Norma & Patrick: Try the LDS(Mormon) Family History Center in your community. Check the Soundex for variations of spellings of your family surname. On St. Thomas the surname LINDQVIST is well known.
When Denmark wished to settle her Caribbean dominion, to ecourage settlers the awarding of land grants with tax incentives was rather commonplace. While Denmark was then also ruling Norway, she did not have sufficient population willing to settle her new Islands. The Dutch became Denmark's surrogate colonizers. Denmark's principal reason for purchasing and colonizing the Islands was to allow her to compete with other European nations in the slave trade. Because of its topography, the largest number of plantations and slaves were on the island of St. Croix.
Beginning ca. 1841, early Danish censuses for the Islands may be found on microfilm at an LDS Family History Center. You will want to investigate available census records for both "Free" and "Unfree" inhabitants. Too, you may find a branch of the federal National Archives and Records Service (NARS) in or nearby your community. That is how I found my paternal ancestors. Hope this information is helpful.
Louise Daniel Hutchinson