You'll need a signal generator and meter or scope. Put a test resistor in series, something between 10 and 1000 ohms.

Find Re. It is the DC resistance of the voice coil.
Find Fs. It is the frequency where impedance is highest.

To find mechanical and electrical Q values, the following formulas are used:

Qms = Fs * (Zmax/Re)0.5 / (Fh - Fl)
Qes = Qms / (Zmax/(Re - 1))
Qts = Qms * Qes / (Qms + Qes)

where

Fs is the resonant frequency of the speaker in free air (Hz)
Zmax is the impedance of the speaker at resonance in free air (ohms)
Re is the DC resistance of the voice coil (ohms)
Fh is the frequency above Fs where speaker impedance is (Zmax*Re)0.5
Fl is the frequency below Fs where speaker impedance is (Zmax*Re)0.5

Note: Fl and Fh can also be found at the points where voltage across the test resistor is equal to Vq in the follwing formula:

Vq = (Vmax*Vmin)0.5

where

Vmin is the voltage across the resistor at the speaker's resonant frequency
Vmax is the voltage across the resistor at a frequency far from resonance

To find Vas using the sealed box method, the following formula is used:

Vas = Vb((Fb / Fs)2 - 1)

where

Vb is volume of the sealed cabinet (ft3, m3 or liters)
Fb is the resonant frequency of the speaker in the box (Hz)
Fs is the free-air resonance of the speaker (Hz)