This is Turdus aonalaschkae pallasii, the hermit-thrush which I have heard in Quebec County. Chapman says (Handbook of Birds of Eastern North America) 'it is most at home in secluded woodland and thickety retreats. . . . Its notes are not remarkable for variety or volume, but in purity and sweetness of tone and exquisite modulation they are unequalled.' Its 'water-dripping song' is justly celebrated.
Eliot here is referring to the lines
356) Where the hermit-thrush sings in the pine trees
357) Drip drop drip drop drop drop drop
Other than discovering that the hermit-thrush is a North American bird don't expect much useful information from this note. However, please consider whether Eliot may have included this note as a hint to an autobiographical basis to the poem. The fact that the hermit-thrush is a North American bird could be significant.
The scientific name for the hermit-thrush, Turdus aonalaschkae pallasii, has been changed since 1922. It is reported that Eliot took delight in the similarity between Turdus and the scatalogical turd.
In Eliot's notes to The Waste Land in his Collected Poems, 1909-1962 the reference to Quebec County has been corrected to Quebec Province.
Eliot did own a copy of Chapman and was a bird-watcher in his early years. Read his poem "Cape Ann."
Links to a sound file of the song of the hermit-thrush and pictures and descriptions are available on the Miscellanea page for this line.