If $R$ is a unique factorisation domain, what is meant by two elements $x,y\in R$ being coprime? We decompose them as $x=up_1\dots p_n$ and $y=u'q_1\dots q_r$, such taht these $p_i,q_j$ are prime elements. Does $x,y$ being coprime mean that $p_i\ne q_j$ for each $i,j$?
One definition for a PID is that $a$ and $b$ are called coprime if $Ra+Rb=R$, i.e., they generate coprime ideals. How this is related is explained here:
Since $R$ is a UFD, it is a gcd-domain, so there exists a gcd for all non-zero elements. Then $a$ and $b$ are coprime, iff $gcd(a,b)=1$.