The businesses in question are KFC and Costa Coffee.
The fast-food drive-thru would be built on land between the A16 and the proposed new home of Boston United Football Club, next to the new roundabout.
It would, according to the documents, have a gross floor area (the area inside the perimeter walls) of about 260 sq m and 42 car parking spaces.
The proposed Costa Coffee development would also be based between the A16 and the planned new stadium, again by the new roundabout, but on the opposite side of the road to the KFC restaurant.
The gross floor area in this case would be 167 sq m and there would be 28 car parking spaces.
As well as a new stadium, The Quadrant development also includes plans for about 500 new homes and the first section of a new distributor road.
David Newton, managing director of developers Chestnut Homes, said: “We are delighted that the first planning applications have been submitted for two of the commercial sites at The Quadrant, which shows the confidence that these well known brands have not only in Boston, but also The Quadrant development.
“Discussions are on-going with other interested parties, and we hope that further applications will come forward during 2018.
“Work is also progressing well on the new homes on the development, and announcements will be made shortly about the show home launch.”