Yes, inbreeding does occur in swans and birds in general. When mammals inbreed, there is a greater chance of genetic abnormalities such as extra or lack of limbs, etc.

However, in swans (birds), these genetic problems do not usually occur and if they do, they show up many many generations later. If there is a genetic anomaly in swans, they usually show up as a color difference or inability to produce viable offspring. The Regal Swan