The Royal Coat of Arms visitors see today in the Council Chamber is a copy of the original. With the Evacuation of the British in 1776, Loyalists took the original to New York and then to Canada.
How the Royal Coat of Arms Arrived at Trinity ChurchEdit
Troops carried the Coat of Arms to New York in March of 1776. In 1783, when the British evacuated New York, a loyalist named Edward Winslow brought the lion and unicorn to his new home in Halifax, Canada. In 1785 Winslow sent the royal emblem to his friend, Judge Ward Chipman, in New Brunswick.  Chipman displayed the Coat of Arms in the court room in St. John, New Brunswick, and then in 1791 he gifted it to Trinity church in St. John. The church placed the Coat of Arms over the Governor's pew, appropriate given that it had once been in the Royal Governor's chamber in the Massachusetts colony. Although a major fire destroyed the church in 1877, the Coat of Arms was saved. Visitors today to St. John, New Brunswick will see it on display at the new Trinity Church. 
When Edward Winslow sent the Royal Coat of Arms to New Brunswick he enclosed a letter. Below is an excerpt from that letter:
In the box with the stationery is a venerable Coat of Arms, which I authorize you to present to the Council Chamber, or any other noticeable public room, which you will think best entitled to it. They (Lyon and Unicorn) were constant members of the Council at Boston…ran away when the others did, have suffered, are of course Refugees, and have a claim for residence at New Brunswick. 
Ed Edward Winslow, Jr. (1746-1815)
Winslow was raised in Plymouth and then attended Harvard, graduating in 1765. After college he returned to Plymouth. While most in the Plymouth Town Meeting radically opposed the Royal Governor, Winslow and his father showed open support for the governor. Winslow even formed a "company of tories" that opposed the revolutionaries.
By 1774, Winslow's politics made it unsafe for him to stay in Plymouth, and he fled to Boston. In 1775 he fought on the side of the British in the Battle of Lexington and Concord. According to some accounts, the horse he rode was shot out from underneath him during the battle. Later he would be appointed “Muster Master General” for the British. In this position his task was to count the Loyalist regiments. His assistant was the deputy Ward Chipman, the man who later received the Royal Coat of Arms from Winslow in New Brunswick.
Winslow moved to Halifax with the evacaution of the British. He would marry a woman named Mary Symonds without the knowledge of his parents or siblings. It is likely he did not want his family to know he had married a woman of a lower social status. They found out about his marriage years later when they moved from New York to join him in Halifax. He already had three children by that time.
Winslow led a political campaign to start a the new province of New Brunswick as a home for loyalist refugees . He served in several government positions in New Brunswick, including as a member of the governor's council.
 Alfred Stockton, ed. The Judges of New Brunswick and their Times. Cambridge, MA: Harvard Law Library, 1915: 123.
 Stockton.The Judges of New Brunswick and their Times. 123.
 Margaret Conrad and Lisa Charlong,“The Winslows: Edward Winslow” Atlantic Canada Virtual Archives, 2005. http://atlanticportal.hil.unb.ca/acva/en/winslow/family/biography.php