Instead of wasting time searching for and spending money buying a watch that has a silent mode, just spend a couple minutes with a tiny screwdriver and some tape to create your own silenced watch.
Method works with pretty much any digital watch, but I seriously wouldn't recommend doing this on a G-shock or anything expensive. You can buy a watch like the one I'm using off of Amazon for like 10 bucks max, and silence that one using this guide.
Comes in handy for AP exams or SAT's (they require you to have a watch without an audible alarm).
The author of this instructable takes no responsibility for any damage caused to you or your property resulting from following this tutorial.
That being said, this is the simplest method I've found and this has been working for me for over two years with no problems : ).
Step 1: Flip the Watch Over.
You'll see the screws that hold the back plate on (there are usually 4 screws).
Step 2: Remove the Screws Holding the Backplate On.
Step 3: We Will Be Working Only With the Backplate.
Step 4: Flip the Backplate Over.
You'll notice two exposed metal rectangles on the top and the bottom. These are the electrical contacts for something called a piezoelectric element (its the little annoying thing that makes noise when u press buttons on your watch)
Step 5: Your Backplate and Piezo Element May Look Something Like This.
No matter what it looks like, to silence the watch you just have to prevent the electrical signal from the watch from reaching this thing.
Step 6: If You Want to Know How the Alarm Works, Take a Look at the Actual Watch for a Second.
You'll notice a protruding spring or pin. When the backplate is on, this spring or pin contacts the piezo element and sends electricity to it when the watch wants to make a sound. To silence the watch, you just have to prevent electricity from reaching the piezo element. The easiest way to do this would be to put something that doesn't conduct electricity between the piezo and the spring or pin.
Step 7: Tape Over the Piezo Element
Tape over any part of the piezo element that is exposed. Tape over the whole backplate if you need to (just don't tape outside the caved in part of the backplate or the backplate won't fit back on the watch well). Taping over the piezo prevents it from contacting the tiny spring or pin, therefore preventing it from making any noise.
Step 8: Put the Backplate Back on and Screw It In.
You might want to put the screws in tightly to make sure that the watch remains waterproof.
Step 9: Fin.
If ever in the future you want to un-silence it, just take the watch apart, remove the tape, and put it back together again.
You now have a silenced watch, and you didn't spend anything extra to make it shut up : ).