There is no electron in the nucleus of an atom. how then the beta particles are emitted from the nucleus of an atom?

In radioactive emissions of beta particles (energetic electrons), a bound neutron in the nucleus changes into a proton inside the nucleus, with the emission of a electron and an anti-neutrino.

Since mass of a free neutron is more than that of a free proton, a free neutron also decays in a similar fashion, i.e into a proton, electron, and anti-neutrino.

A bound proton also changes into a neutron with the emission of a positron, and a neutrino. It needs to be emphasized that only a bound proton inside the nucleus, undergoes this conversion i.e. into a neutron, a positron, and a neutrino.

So although, electrons (positrons) are not present in the nucleus, these are spontaneously created at the time of decay and are emitted.