No, you don't need to believe in God,or even to follow the way of Buddha, in order practise Mindfulness successfully.
The practice did evolve, though, from Buddhist and Hindu traditions, and you may find it helpful to understand some basics of those traditions. The key Buddhist concept you need to understand is dukkha, which should be recognisable to any busy teacher:dukkha rolls up the English language constructs of stress, anxiety, dissatisfaction, frustration, unease and discontentment. You don't have to follow the Buddhist path in order to appreciate that any practical technique aimed at overcoming dukkha is worthy of attention.
Buddhists believe that all human life is linked with dukkha and that stress, anxiety, dissatisfaction and discontentment inevitably represent a component of our experience. This is not intended to be a pessimistic view; rather a practical and realistic view of the human condition.
Buddhists believe that there is a path, involving meditation, which if followed, will either ease or cease the dukkha effects.
Another important Buddhist concept is upādāna (clinging, grasping, attachment) and taṇhā (craving, desire). It is largely though a process of letting-go - letting go of greeds, grudges and griefs - that meditation is effective.
Some tetchy teachers will notice that I'm using lots of foreign words with simpler English words in brackets; the reason I don't just use the English equivalents is that they don't represent a 100% match, and understanding these sometimes waffly concepts is an important element in the journey.
However, as I said at the beginning: you don't need to believe in a God, or show and interest in Buddhist tradition, or even use funny foreign words; the research shows that it tends to work anyway.
Meditation, in fact, appears in virtually all religious traditions, but it is a Buddhist variation (Vipassana, for vocabulary freaks) which is most frequently adopted and adapted in psychology and other disciplines to achieve mundane and worldly benefit. Buddhist meditation techniques are employed by psychologists and psychiatrists to help with a whole range of health conditions such as anxiety and depression.