What is the dipole moment for azulene?

The dipole moment of azulene is ##color(blue)("1.08 D")## or ##color(blue)(3.60×10^(-30) "C·m")##.

In the various resonance structures for azulene, the most stable contributors put negative charge in the 5-membered ring and positive charge in the 7-membered ring.

(from commons.wikimedia.org)

In effect, this gives an aromatic 6 π-electron five-membered ring fused to an aromatic 6 π-electron seven-membered ring, as we see in the above diagram.

This separation of positive and negative charge gives the molecule a dipole moment of ##"1.08 D"##.