From my understanding no, not elemental mercury. The main danger with that is inhaling mercury vapor.

However the broken thermometer could introduce open wounds which could do something I’m not sure.

Here’s a guy swishing some around his mouth, he explains more about how the body generally can’t absorb elemental mercury. Not through skin or mucus membranes. youtu.be/DNpdMz0Cfv0

He’s been tested and has not shown to have unusually high levels of mercury in his blood.