Yes, you can compost strawberries.
They rot down quickly in a compost heap – they’re a “green” (even when they’re red ;)), so if you’ve got a lot to get rid of, add them alongside “browns” (like newspaper, egg boxes or straw) to help soak up/take advantage of the excess moisture.
If you’re growing your own, some people say strawberry plants should be replaced every three years because they produce less fruit after that. You can compost strawberry plants – but remember to peg down & pot on, or at the least, just snip off the runners first though. Those new runners will want to grow – better to have them to replace their parent plant in your strawberry bed than letting them use your compost heap as their new home.