In this paper, we first introduce the quermassintegrals for convex hypersurfaces with capillary boundary in the unit Euclidean ball
B
n
+
1
{\mathbb {B}}^{n+1}
and derive its first variational formula. Then by using a locally constrained nonlinear curvature flow, which preserves the
n
n
-th quermassintegral and non-decreases the
k
k
-th quermassintegral, we obtain the Alexandrov-Fenchel inequality for convex hypersurfaces with capillary boundary in
B
n
+
1
{\mathbb {B}}^{n+1}
. This generalizes the result of Scheuer [J. Differential Geom. 120 (2022), pp. 345–373] for convex hypersurfaces with free boundary in
B
n
+
1
{\mathbb {B}}^{n+1}
.