No, it is not possible for a hydrogen $({}^{1}_{1}\mathrm{H})$ nucleus to emit an alpha particle $({}^{4}_{2}\mathrm{He})$. It is because it would mean that the mass number of the daughter nucleus $({}^{4}_{2}\mathrm{He})$ is more than that of the parent nucleus $({}^{1}_{1}\mathrm{H})$ which is not possible.