Yes, Dr. Henry Taylor was - if not a Quaker himself - certainly sympathetic; there were Quaker meetings at his house in Flushing and his second wife, Sarah Palmer, was a Quaker. I do have the info re Nicholas Davis but no actual documentation on it. But it sounds right.
here is the part of my Taylor tree that includes your John:
1Dr. Henry TAYLOR1630/45?, will dated 5 Jun 1718, proved 19 May 1719, arr. Boston?+ 1) Mary, m.c1664 Boston; 2) Sarah Palmer (Quaker - her brother is John Palmer), m.c1686 (when he is grantor, with Sarah, of property in Flushing to John Harrison). Sarah married 2) Benjamin Field (b.1668? Flushing), on 13 Apr 1727
2Hannah 7 Jul 1665, d.young?
2John 1666, d.young?
2Mary 1669, d.young?
2Henry Jr. c1670, d.young?
2John, 4 Aug 1674Oyster Bay + 1) Catherina Louwerse (had son James, bapt.18 Oct 1705); 2) Mary Whitehead, 17 Jun 1681, d.Woodbridge, Middlesex Co., NJ. After John’s death, she married 2) Thomas Burroughs, d.1703; 3) c1703 Rev. William Urquhart c1675 England - 1707 Jamaica LI NY.(Mary Whitehead was dau ofMajor David Whitehead, 1646-1704 + Abigail Stephenson)
3James, bapt.Oct.18, 1705. Ref.Dutch Church(of Catherina Louwerse)
3Abigail 1695 + Rev. Benjamin Woolsey (of Mary Whitehead)
2Mary 17 Apr 1678 - 17 Apr 1769 + Colonel Francis Willett 29 Jun 1693 – 6 Oct 1776 (son of Andrew Willett, merchant of Rhode Islandand Ann Coddington; and grandson of Capt. Thomas Willett, the first English speaking Mayor of NYC, in 1664); m.c1715 (no children). [See Parkinson Tree]
2Sarah 12 May 1688+ William Marsh, m.c1711
3William Marsh Jr.
2Phebe 9 Mar 1690/1, d.1714 + William Doughty c1676; m.c1709, Flushing
2Benjamin TAYLOR c1676 - >14 Mar 1742 + Arianthe BERGEN (widow of Jan Garretszen or Garrason; she had 4 children by him), m.26 May 1714, at Grace Church, Jamaica (by Rev.Th.Poyer)