Main Difference
The main difference between Hexagon and Quadrilateral is that the Hexagon is a polygon with six sides and Quadrilateral is a polygon with four sides.

Hexagon
In geometry, a hexagon (from Greek ἕξ hex, “six” and γωνία, gonía, “corner, angle”) is a six sided polygon or 6gon. The total of the internal angles of any hexagon is 720°.

Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5sided), hexagon (6sided) and so on.
The origin of the word “quadrilateral” is the two Latin words quadri, a variant of four, and latus, meaning “side”.
Quadrilaterals are simple (not selfintersecting) or complex (selfintersecting), also called crossed. Simple quadrilaterals are either convex or concave.
The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is
∠
A
+
∠
B
+
∠
C
+
∠
D
=
360
∘
.
{displaystyle angle A+angle B+angle C+angle D=360^{circ }.}
This is a special case of the ngon interior angle sum formula (n − 2) × 180°.
All nonselfcrossing quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

Hexagon (noun)
A polygon with six sides and six angles.

Quadrilateral (noun)
A polygon with four sides.

Quadrilateral (noun)
An area defended by four fortresses supporting each other.
“The Venetian quadrilateral comprised Mantua, Peschiera, Verona, and Legnano.”

Quadrilateral (adjective)
having four sides.

Hexagon (noun)
a plane figure with six straight sides and angles.