The reference I found in the History of St. Clair Illinois alludes to a kinship relationship. Palma Scheumack did not believe there was a blood relationship. Reynolds'cousins, i.e. Hugh Forgey's children, were cousins of John Beaird's children. Sort of a convoluted relationship :D.
"John Beaird was a prominent citizen of Knox County, Tennessee, which county he represented in the Tennessee Legislature. He was usually selected as leader when the community had trouble with the Indians. He was brave, energetic, and a successful Indian fighter. He came with his family to Illinois in 1801 and settled four miles from Kaskaskia. His son, Joseph, lived for some years in Cahokia, and was a member of the Legislature for several terms. The family was related to that of Governor Reynolds, and when John Reynolds, afterward Governor, started out to practice law, he made his home for a time with Joseph Beaird, at Cahokia. William A. Beaird, another son of John Beaird, naturally possessed a good mind, but obstinately refused to get an education or to receive any information thru the medium of books. However, by observation and experience, he did acquire much practical knowledtge. He never married. He served as Sheriff of St. Clair County from 1815 to 1830. He died in Belleville in 1843."