Can an electron make quadrupole gravitational waves? A gravitational wave is a quadrupole wave. Now when an electron is accelerated it usually emits a photon. But can an electron also emits an gravitational wave? If so how does it 'make' an quadrupole wave?
 A: To create gravitational waves, you need objects (with asymmetric mass) whose motion involves acceleration and its change, provided the motion is not spherically or rotationally symmetric.

A simple example of this principle is a spinning dumbbell. If the dumbbell spins around its axis of symmetry, it will not radiate gravitational waves; if it tumbles end over end, as in the case of two planets orbiting each other, it will radiate gravitational waves.
The heavier the dumbbell, and the faster it tumbles, the greater is the gravitational radiation it will give off. In an extreme case, such as when the two weights of the dumbbell are massive stars like neutron stars or black holes, orbiting each other quickly, then significant amounts of gravitational radiation would be given off.

https://en.wikipedia.org/wiki/Gravitational_wave
More technically, the second time derivative of the quadrupole moment of an isolated system's stress-energy tensor must be non-zero to create gravitational waves.

Any EM radiation influences the gravitational potential on it‘s path (as well as any mass do). Sending periodically EM radiation (radio waves with its hugh number of photons, sended periodicaly by the wave generator) will change periodically the gravitational potential and by this we get a gravitational wave. Not detectable with our possibilities now.

Does the oscillating electric and magnetic field of a photon generate gravitational waves?
Now in your case, as an electron (defined in the SM as an elementary point particle) alone (as an isolated system) does not satisfy this, it would not emit gravitational waves. To create gravitational waves that are detectable at our current level of capabilities, you need a system of two objects, like two black holes, merging.
To create gravitational waves you need:

*

*an asymmetric mass (stress-energy) distribution in your system


*relative motion that cannot be spherically or rotationally symmetric
An accelerated electron (treated as an isolated system, a point particle) fails in 1., since there is no asymmetric mass (stress-energy) distribution.
If we take the electron together with the external field (that provides the energy for the acceleration), and we say for the sake of argument that 1. is satisfied (having an asymmetric stress-energy distribution between the electron and the external field), then the system fails in 2., for not having a asymmetric relative motion (because the electron moves for example in the antenna back and forth symmetrically).
A: The answer by Physics Guy says:

Since the electron moves in spacetime and has mass, it produces gravitational waves. that can be derived from General Relativity.

This is wrong. Motion and mass are not sufficient criteria.
The answer by Árpád Szendrei says:

Now in your case, as an electron alone (as an isolated system) does not satisfy this, it would not emit gravitational waves. To create gravitational waves that are detectable at our current level of capabilities, you need a system of two objects, like two black holes, merging.

This is also wrong. If the electron is accelerated, then it is not an isolated system.
In general, the criterion for an object to emit gravitational waves is that its mass quadrupole moment has to vary with time. In most cases, an accelerated electron would satisfy this criterion, so the answer to the question is yes.
A: Since the electron moves in spacetime and has mass, it produces gravitational waves. that can be derived from General Relativity. 
