Does there exist a probability measure on the integers such that, the probability of any two arithmetic progressions with the same difference part, is the same?
We assume the probability measure is defined over the power set of the integers. Hence, a probability measure corresponds to a sequence of positive (non-negative) numbers $\{p_z\}_{z \in \mathbb{Z}}$ which sum up to one.
If I understood well, denoting $p$ this probability measure defined on $\mathcal P(\mathbb Z)$, you want that for all $a,b,c\in\mathbb Z$, $p(a+c\mathbb Z)=p(b+c\mathbb Z)$?
I'm afraid such a probability measure does not exist.
Indeed, let $x\in\mathbb Z$. For all $n\in\mathbb N^*$, you have $$ p(n\mathbb Z)=p(1+n\mathbb Z)=\cdots=p(n-1+n\mathbb Z), $$ and $$ p(n\mathbb Z)+p(1+n\mathbb Z)+\cdots+p(n-1+n\mathbb Z)=p(\mathbb Z)=1, $$ hence $$ p(x+n\mathbb Z)=\frac1n\cdot $$
We deduce that $$ p(\{x\})\le p(x+n\mathbb Z)\le\frac1n\underset{n\to+\infty}{\longrightarrow}0, $$ and therefore $p(\{x\})=0$ for all $x\in\mathbb Z$, which implies $p(\mathbb Z)=0$.