Based on the wiki's coverage, Caleb Williams did not get injured in the Bears vs. Patriots game — in fact, he played well.
The Bears defeated New England 5-2 on Thursday Night Football in Week 7 (Oct 23), and the wiki notes that Williams "maintained his strong performance since returning from injury" in that game (from the Chicago Bears page). The injury he was dealing with was a torn hamstring suffered back in Week 2 against Minnesota, which cost him Weeks 3–5. He returned in Week 6 vs. Atlanta (30/36, 305 yards, 1 TD) and carried that momentum into the Patriots game without any new injury issues.