Yes, you have several options. The easiest is likely to be purchase a **1000baseT SFP** for your "fiber switch," which allows you to plug a gigabit copper connection into it. You probably already know there are various single-mode and multi-mode fiber SFP transceivers, and there are copper ones available as well. These are not guaranteed to work in every switch, so, check with your manufacturer or test it out in a lab.
There are also **media converters** which do what you describe; convert from one type of media (like copper 1000baseT) to another (like 1000baseLX fiber.) These devices come with plenty of gotchas (like port up/down state issues, masking errors, and other things that make malfunctions hard to troubleshoot), so don't use a media converter unless you don't have a better alternative.