Suppose we want to test whether $Z$ is a confounding variable for the effect of $X$ on $Y$. Is it enough to just check unadjusted and adjusted estimates of the coefficient of $X$ and see if they differ? Also, if the difference between the two is very small (e.g $1/1000$) can we say that there is no confounding? It seems in survival analysis it would be complicated to check for the association of $Z$ vs. $X$ and $Z$ vs. $Y$.
First, it isn't terribly complicated to check for the association of Z on X or Y, even in survival analysis. Propensity scores and Inverse Probability of Treatment Weights (both common methods for adjusting for confounding in a survival context), along with other somewhat more esoteric methods are based on estimating the relationship between the covariate and the exposure or outcome.
You can compare the adjusted and unadjusted score to evaluate whether or not there is confounding, but only if you have reason to believe there's confounding there in the first place. Just a raw comparison of the adjusted and unadjusted estimates run the risk of having actually induced confounding by adjusting for a variable that is affected by both the exposure and the outcome. Check up on the literature on directed acyclic graphs for covariate selection, and read about "colliders" for an explanation of this phenomena.
But once something is believed to be a confounder for any number of reasons - subject matter expertise, the use of a DAG, establishing that the variable meets the criteria for a confounder, one can use what you're suggesting - which is normally called a change-in-estimate approach - to check whether or not its a "problem", based on how much the estimate changes. The threshold for what is a problem varies, but in Epidemiology, it's often a 10% change in estimate that's used to say something confounds the exposure-disease relationship enough to be worth adjusting for.
There is no test for confounding, unfortunately. If you think Z is a confounder, adjust for it.
It's not sufficient to look at changes (of size 10% or otherwise) because, even when Z isn't a confounder, when using non-collapsible measures of effect - like odds ratios - the parameter estimated in the adjusted analysis is different from that estimated in the unadjusted analysis.