Victoria served as the queen of the United Kingdom of Great Britain and Ireland from 1837 until 1901.
As far as is known, Queen Victoria never met Whistler. When he was President of the Society of British Artists, he petitioned her for permission to rename the Society the Royal Society of British Artists, which was granted (see #05969, #13568, #05971, #05973). In thanks, he presented her with an album containing a set of etchings, 'Naval Review set', 1887 (K. 316-331) (excat 7) (see #05977). Their correspondence was were always funnelled through official channels.
In addition, a number of his etchings were bought for the Royal Collection as Windsor Castle, but this probably reflects the taste of the curators of the collection rather than Victoria herself. See for instance, Whistler's letter to Richard Holmes regarding purchases in 1878/1879 (#11306).
Dictionary of National Biography, Oxford, 2004.