There is currently no data indicating that cucumbers have toxic potential to pets, Julie. However, it is important to keep in mind that even vegetables or plants considered to be nontoxic can produce minor stomach upset if ingested.
While the ASPCA typically does not advocate the feeding of table foods to pets, if you choose to offer your dog a bit of cucumber now and then, be sure to keep an eye out for signs of gastrointestinal (GI) upset; if you notice any symptoms, it’s a good idea to discontinue giving him this veggie.