It is unlikely. Jupiter’s orbital period is 11.86 years, whereas the sunspot cycle averages out to 11.05 years. And the sunspots are actually representative of variations in the Sun’s magnetic field, which because it changes in polarity actually has a full cycle of 22 years.
It is also believed to be generated by events deep in the Sun, so it is difficult to conceive of any physical force from Jupiter that would be strong enough to drive it.
If we assume that Jupiter's effect is gravitational, then it somehow has an impact on the magnetic environment on the sun. Let's hypothesize that the effect is to slightly increase the magnetic field magnitude.
This increase in magnitude would happen regardless of whether the polarity of the field is positive or negative, thus introducing a perturbation with a period of roughly 11 years. The effect on sunspots, for example, may not depend upon the polarity of the field.
Now let's turn to the slight difference between 11.05 and 11.86. I will describe a mechanism that I built as part of a lab experiment.
Consider a cylindrical mass suspended from a vertical spring with the axis of the cylinder aligned with the axis of the spring. If one displaces the spring downward, for example, the spring will stretch, storing potential energy.
If one then releases the spring, the spring force will accelerate the mass upward until it reaches the original equilibrium point. The mass will then continue upward until the compression of the spring decelerates the mass to a halt. Then the compressed spring accelerates the mass downward until the mass passes the equilibrium point. At that point, the spring starts being stretched and decelerates the mass until the mass reaches the point where it was originally released.
The period of the spring mass system just described can be calculated if one knows the mass and the spring constant.
Now we complicate things a bit.
Instead of initially pulling the mass down, we rotate it around its axis. This will wind up the spring and create a restoring force that can rotate the mass back to its equilibrium point. If we choose the diameter of the cylindrical mass and the stiffness of the spring just right, we can get the frequency of the oscillatory behavior of the torsional spring mass system to be quite close to the frequency of the vertical spring mass system.
For example, lets choose a mass and spring constant that results in a vertical oscillation of 12 cycles per second. Let's choose the diameter of the mass and the stiffness of the spring to get a torsional oscillation of 11 cycles per second.
What I have just described is a system of two, coupled oscillators consisting of one spring and one mass.
If we set this system oscillation by pulling downward on the mass and releasing it, we will see the result of the coupling. Initially, the system behaves as a vertical oscillator. But each time the mass moves up or down, it twists the spring a little bit. Eventually, all of the energy put into the vertical oscillator will be translated to the torsional oscillator.
At this point, each time the mass rotates clockwise or counterclockwise, it stretches or compresses the spring a little bit. Eventually, all of the energy that flowed into the torsional oscillator will return to the vertical oscillator.
The rate at which the energy moves back and forth between the two oscillators is called the beat frequency. It is the difference between the two frequencies of oscillation. In this example, the energy transfer would occur once per second.
The actual system I built had frequencies more like 1 cycle per second for one system and 1.1 cycle per second for the other. The beat frequency is then 0.1 cycle per second. This resulted in seeing the energy move from one system to the other and back again every ten seconds.
The point of describing this system is that the fact that the two frequencies differ does not mean that they cannot interact.