Two minutes before the break, Yeovil went close as Akpa Akpro's goalbound effort was superbly blocked by Dean Wells.
But the forward did get his name on the scoresheet on the stroke of half-time as he volleyed home Otis Khan's cross.
Stevenage were level eight minutes after half-time when McQuoid tapped home from close-range after a corner was not cleared.
Boro were on top and Steve Schumacher then forced Jonny Maddison into a good low save before the Yeovil goalkeeper parried Jobi McAnuff's powerful drive.
At the other end, Akpro robbed Jack King of the ball before squaring it to Francois Zoko, but the striker's flicked finish crept agonisingly wide.
Tom Pett's shot was parried as Stevenage searched for a winner.
With seven minutes left, Maddison made another crucial stop from substitute Jamie Gray after he was put through one-on-one, but Stevenage had to settle for a draw which leaves them one place and one point outside the play-offs.
Match report supplied by the Press Association
Match ends, Yeovil Town 1, Stevenage 1.
Second Half ends, Yeovil Town 1, Stevenage 1.
Foul by Matthew Dolan (Yeovil Town).
Tom Pett (Stevenage) wins a free kick in the attacking half.
Substitution, Yeovil Town. Matt Butcher replaces Ben Whitfield because of an injury.
Delay over. They are ready to continue.
Attempt saved. Francois Zoko (Yeovil Town) right footed shot from the left side of the box is saved in the centre of the goal.
Corner,  Stevenage. Conceded by Jonny Maddison.
(Yeovil Town) wins a free kick in the attacking half.
Foul by Dean Wells (Stevenage).
Corner,  Stevenage. Conceded by Jonny Maddison.
Attempt saved. Jamie Gray (Stevenage) left footed shot from the centre of the box is saved in the bottom left corner.
Ben Whitfield (Yeovil Town) wins a free kick on the right wing.
Foul by Connor Ogilvie (Stevenage).
Substitution, Stevenage. Luke Wilkinson replaces Josh McQuoid.
Corner,  Yeovil Town. Conceded by Jack King.
Foul by Alex Lawless (Yeovil Town).
Michael Tonge (Stevenage) wins a free kick in the attacking half.
Attempt saved. Jean-Louis Akpa-Akpro (Yeovil Town) header from the left side of the six yard box is saved in the top centre of the goal.
Corner,  Yeovil Town. Conceded by Connor Ogilvie.
Substitution, Stevenage. Jamie Gray replaces Charlie Lee.
Attempt missed. Michael Tonge (Stevenage) header from the right side of the six yard box is close, but misses the top right corner.
Corner,  Yeovil Town. Conceded by Dean Wells.
Jean-Louis Akpa-Akpro (Yeovil Town) wins a free kick in the defensive half.
Foul by Ronnie Henry (Stevenage).
Substitution, Stevenage. Michael Tonge replaces Ben Kennedy because of an injury.
Delay in match  (Yeovil Town).
Attempt missed. Francois Zoko (Yeovil Town) right footed shot from the centre of the box is close, but misses to the left.
Corner,  Stevenage. Conceded by Jonny Maddison.
Attempt saved. Ben Kennedy (Stevenage) left footed shot from the centre of the box is saved in the top centre of the goal.
Attempt saved. Jobi McAnuff (Stevenage) right footed shot from the left side of the box is saved in the centre of the goal.
Foul by Francois Zoko (Yeovil Town).
Connor Ogilvie (Stevenage) wins a free kick in the defensive half.
Attempt missed. Jack King (Stevenage) header from the right side of the six yard box is just a bit too high following a corner.
Corner,  Stevenage. Conceded by Jonny Maddison.
Attempt saved. Steven Schumacher (Stevenage) right footed shot from outside the box is saved in the bottom left corner.
Matthew Dolan (Yeovil Town) wins a free kick in the defensive half.
Foul by Ronnie Henry (Stevenage).
Goal!  Yeovil Town 1, Stevenage 1. Josh McQuoid (Stevenage) right footed shot from very close range to the bottom left corner. Assisted by Jack King.
Corner,  Stevenage. Conceded by Ryan Dickson.
Josh McQuoid's close-range finish earned play-off chasing Stevenage a point in an end-to-end game at Yeovil.