Sir Gilbert Elliot was Member of Parliament for Selkirkshire between 1753 - 1765
He was a Tory Member of the King’s Privy Council in 1762
He was Member of Parliament Roxburghshire between 1765 - 1777
He succeeded to the title of 3rd Baronet Elliot of Headshaw in 1766
He was Treasurer of the Royal Navy 1770 - 1777
Sir Gilbert Elliot married Agnes Murray Kynynmound, heiress of Melgund (Forfar, Scotland) and heiress of Lochgelly and Kynynmound (Fifeshire, Scotland)
children - Isabella, Gilbert, Hugh, Alexander, Robert, David & Eleanor
from “Border Memories” by Walter Riddell Carre
Sir Gilbert, third baronet, carried on the line of the Minto family. Like his father and grandfather, he went to the Bar, and passed advocate in 1743. He did not follow up the profession, preferring to devote himself to politics, in which he was as successful as his predecessors were in jurisprudence. Sir Gilbert first represented Selkirkshire, and was afterwards elected for Roxburghshire — his Parliamentary career continuing for about twenty-three years, when his health broke down, and he died at the comparatively early age of fifty-four. Sir Gilbert held several important appointments, and was a Privy Councillor, standing very high in general estimation. He had good literary taste, and was an accomplished speaker. He was also a good writer of verses, having commenced versifying at he very good pastoral one beginning " My sheep I neglected," &c. Sir Gilbert had the honour of holding several important appointments, and he was a member of the Poker" Club, as well known then as the New Club is now. It was instituted in 1762, and continued for several years, being supported by some of Scotland's brightest stars, then brilliantly shining, such as David Hume, Adam Smith, and Adam Fergusson, who, I believe, invented its name, enigmatical to the public, though it had something to do with the militia question, the poker being an instrument for stirring up. ￼
Border Memories by Walter Riddell Carre Esq. of Cavers Carke 
The Peerage of the United Kingdom of Great Britain & Ireland v1 by John Debrett 
It may be possible to confirm family relationships with Gilbert by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share DNA with Gilbert: