Unless it is a very small boat, it would noticeably increase the submarine's drag.
Which they would notice...eventually...maybe.
Here we have equipment that shows speed through the water.
Here we have an indication of shaft RPM.
Yes there is a correlation between these two things.
No, it is not a simple linear relationship, nor is it a constant, nor is it something that is monitored as a high priority.
Someone will no doubt eventually say "Man, do turns seem really high for this speed at this depth?" People will start puzzling over this. "Water temp?" "Salinity?" "Are the indications correct?" That one by the way will take some time to get past the probably not stage, because unless things have changed substantially speed through the water is a really crappy system that nobody really cares about that much to start with so it is always taken with a grain of salt. Somebody will be checking movement by the plot, and the actual priority is going to be making sure we really are where we think that we are.
Then there's a bit of WTH? Eventually, maybe, someone says "Could we be snagged and dragging?" About then sonar is starting to think it's strange that some fishing boat has coincidentally steered onto our track. Not that that doesn't happen. There really are only so many directions to go, and if you are snagging a fishing boat you are probably not out in the wide open ocean so that reduces the possibilities.
BUT...all this has to happen faster than their nets give way. Or get cut away. Which if the fishing boat people are smart they are doing with some alacrity, because as soon as they know they have a submarine they need to be thinking "It can go deep...and it can come back. We can't."