Identify the complex number expression.

Let the given expression be $W$.

$$W = \frac{Z-1-i}{3+2i}$$

The conjugate of a quotient is the quotient of the conjugates: $\left(\frac{A}{B}\right)^* = \frac{A^*}{B^*}$.
So, the conjugate of $W$, denoted as $W^*$, is:

$$W^* = \frac{(Z-1-i)^*}{(3+2i)^*}$$

The conjugate of a sum or difference is the sum or difference of the conjugates. The conjugate of a real number is itself, and the conjugate of $i$ is $-i$.

$$(Z-1-i)^* = Z^* - 1^* - i^* = Z^* - 1 - (-i) = Z^* - 1 + i$$

The conjugate of $a+bi$ is $a-bi$.

$$(3+2i)^* = 3 - 2i$$

Multiply the numerator and the denominator by the conjugate of the denominator, which is $3+2i$.

$$W^* = \frac{Z^* - 1 + i}{3 - 2i} \times \frac{3 + 2i}{3 + 2i}$$

$$(Z^* - 1 + i)(3 + 2i) = Z^*(3) + Z^*(2i) - 1(3) - 1(2i) + i(3) + i(2i)$$ $$= 3Z^* + 2iZ^* - 3 - 2i + 3i + 2i^2$$ $$= 3Z^* + 2iZ^* - 3 + i + 2(-1)$$ $$= 3Z^* + 2iZ^* - 3 + i - 2$$ $$= 3Z^* + 2iZ^* - 5 + i$$

$$W^* = \frac{3Z^* + 2iZ^* - 5 + i}{13}$$

The conjugate of the expression is:

$$\frac{3Z^* + 2iZ^* - 5 + i}{13}$$