The general answer is sometimes Yes, various unfold/refold pairs
of convex polyhedra exist, but not always,
depending on what is meant by "cut."
Here is an example that Erik Demaine, Marty Demaine, Anna Lubiw, and I worked out carefully:
The Latin-cross unfolding of the cube can refold
into precisely 23 distinct convex polyhedra,
as displayed below (all of the same surface area):
two doubly covered flat quadrilaterals,
three pentahedra, each with one or more quadrilateral faces,
and six octahedra:
Fig. 25.30, p.408 in Geometric Folding Algorithms: Linkages, Origami, Polyhedra, 2007.
Here is a "movie" of one of the refoldings to a tetrahedron:
But more generally, in the 2012 paper Refold Rigidity of Convex Polyhedra
, we showed that every
polyhedron can be cut open and refolded to an incongruent convex polyhedron.
But if the cutting is restricted to follow edges of the convex polyhedron
then there are "refold-rigid" polyhedra.
For example, each
of the 43,380 edge unfoldings of a dodecahedron may only fold back to
I apologize for not initially understanding Morteza's precise question, which
asks whether or not, for any two given pair of convex polyhedra $P$ and $Q$
with the same surface area, is it always possible to cut
open $P$ and refold to $Q$. This remains an open question as far as I know.
It is a version of Open Problem 25.31 (Fold/Refold Dissections) in
Geometric Folding Algorithms