No, I don't believe we are related to this Baird line.
William Baird (1704-1793), who had four sons in the Revolution, was the son of Alexander Baird. Alexander came over from Scotland, was a lawyer in Bushwick, Kings Co., NY (present-day Brooklyn) and owned a farm in Griggstown, NJ.
However, according to a 1918 article in the Somerset County Historical Quarterly (Vol. 7, page 132), William Baird "may not have been related to John Baird, who heads the Monmouth co. Baird line, but the two lines have always maintained there was a connection." This connection probably goes quite a ways back.