My Italian Greyhound sometimes gets a weepy eye, probably got a little cat hair in it. It usually goes away with in a few hours but I am sure you have to make a vet appointment within a certain time frame if you just got him and if its not gone away by then they may give you some sort of ointment to put on it.
Welcome...I have a Bichon Frise' who is 3 now. Some Bichons get teary eyes...sometimes it can be from a blocked tear duct. Sometimes you can massage the corner of the eye...I had to have Chloe's unblocked by the vet. They did it when she was spayed. I'm sure when you take the pup to the vet for the checkup that you should do with any new puppy, they can give you some idea as to what it is.