For two given graphs
$G_{1}$
and
$G_{2}$
, the planar Ramsey number
$PR(G_{1},G_{2})$
is the smallest integer
$N$
such that every planar graph
$G$
on
$N$
vertices either contains
$G_{1}$
, or its complement contains
$G_{2}$
. Let
$C_{4}$
be a quadrilateral,
$T_{n}$
a tree of order
$n\geq 3$
with maximum degree
$k$
, and
$K_{1,k}$
a star of order
$k+1$
. We show that
$PR(C_{4},T_{n})=\max \{n+1,PR(C_{4},K_{1,k})\}$
. Combining this with a result of Chen et al. [‘All quadrilateral-wheel planar Ramsey numbers’, Graphs Combin.
33 (2017), 335–346] yields exact values of all the quadrilateral-tree planar Ramsey numbers.