Queen Mary doesn't seem to have been fond of horses or dogs and was rarely photographed around one--I don't think I've ever seen her on horseback. She did have at least one family dog when she was a Teck Princess and was around them when she married George, but I don't think she herself owned one.
""Queen Mary was terrified of dogs, and lived in fear of the ferocious little Welsh corgis favoured by her granddaughter, the future Elizabeth II. At a Buckingham Palace garden party, Princess Elizabeth handed her a dog biscuit to give to the most ill-tempered of the corgis. Queen Mary in panic handed it on to her nearest neighbour, the Archbishop of Canterbury who, mistaking it for a teatime titbit, gratefully popped it into his mouth."