The question is stated informally, using the terms "queries" and "access". The way I interpret it, the answer is yes. 

Indeed, let the $j$th query give us the values $g(q_j)$ and $g(q_j+s)$, where $(q_j)_{j=1}^\infty$ is an enumeration of the set of all rational numbers in $[0,1]$. Suppose that the set 
$$E:=\{x\in[0,1]\colon x+s\in[0,1],g(x+s)-g(x)<t\}$$
is nonempty. Since $g$ is continuous, the set $E$ is open in $[0,1]$, and hence $q_n\in E$ for some natural $n$. So, we will find the point $q_n\in E$ on our $n$th query.