Is it permissible to apply for a full scholarship offered by the government of a Western country, for a study program that requires traveling there?

Eagerness for beneficial knowledge is praiseworthy. If you are secure in your faith and believe you will remain steadfast in obedience to Allah, and the required specialization is not available in Muslim countries, then there is no harm in applying for the scholarship. However, if you fear that you will not uphold the limits of Allah, then avoid it, for faith is more important than worldly matters. Residing in lands of disbelief is not permissible except out of necessity or a recognized need.