Can the SSA3000X Plus, SSA3000X-R, or SVA1000X series analyzer automatically measure the quality factory (Q) of a filter?

Yes.

NOTE: This is not currently available on the SSA3000X series.

The quality factor of a filter is defined as:

Q = Center frequency/Bandwidth

From the SSA3000X Plus, X-R, SVA1000X Series Manual:

 

You can find the automatic Q measurement in the Measurement function menu.

Press the Maker Fn button > Select N dB BW marker type and set the amplitude level (dB) at which you wish to measure the bandwidth (BW).

The measurements will be shown in the upper left-hand-side as shown below: