I have a compost pile from burned brush that I have been adding aspen shavings to (from guinea pig cage) and have kept it tilled. The guinea pigs only eat timothy hay, carrots, cucumbers, roma tomatos, and romaine lettuce. I have had 4 tomato plants and 3 cucumber plants come up on their own. The cucumber plants are huge and are producing what look like gourds instead of the long dark cucumbers we fed them. Are these plants good for vegetable production?
It is not unusual for seeds to germinate in a compost bin.
You can certainly leave the plants to grow there or move them to a garden location.
The seeds most likely were from Hybrid plants, so therefore the new plants may not be the same as the parent plants. This would explain the cucumbers looking 'different'.
If you eat the fruit of these plants wash them well!
Here is a link about this.