1985 - FG - Punt - Fumble - Fumble
1996 - Punt - Int - TD - TD
2001 - Punt - Punt
2003 - Missed FG - Punt
2004 - Punt - Punt - Punt
2007 - First possession went into 2nd Quarter (TD)
2011 - Safety (intention grounding)
2014 - Punt - Interception
2016 - Punt - Punt
2017 - FG
So, there you have it. The Patriots have only scored 3 points in the first quarter of Super Bowls during the Brady/Belichick era. Add in the safety and they are +1 for points by the offense.
In their 13 1st quarter possessions they have had 9 punts, 1 missed field goal, 1 made field goal, an interception, and a safety.
The last time the Patriots had a first quarter Super Bowl Touchdown was when the logos were fun and the Bledsoe led Patriots were coached by the other Bill, Bill Parcells. After punting and throwing a pick, Bledsoe led the team on two touchdown drives to end the quarter.