Yes, the motion of stars and planets can generate gravitational waves, although the effect is typically very small and difficult to detect. Gravitational waves are ripples in the fabric of spacetime caused by the acceleration of massive objects. According to Einstein's theory of general relativity, any accelerating mass can potentially generate gravitational waves.
In the case of stars and planets, their motion can produce gravitational waves if they undergo significant acceleration or experience strong gravitational interactions. However, the magnitude of the gravitational waves generated by individual stars and planets is usually extremely tiny and well below the current detection capabilities of gravitational wave detectors such as LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo.
Gravitational waves from astrophysical sources such as the merger of binary black holes or neutron stars, which involve much larger masses and stronger accelerations, are more likely to produce detectable signals. These events release enormous amounts of energy in the form of gravitational waves and can be observed by sensitive detectors on Earth.