Welcome, Jenny!
Sorry about the delay in answering your question! You can grow the freshly saved seeds, but unless you refrigerate them in wet sand for two months, they will wait until next spring to sprout. Once they sprout, they will generally take 3-7 years to bear, and what you get may not be a greengage, since they hybridize freely with other plums. You might get the next 'Victoria', or you might get a sour "spitter"! Still, worth doing if, like me, you're 'satiably curious, and love growing things.