Yes, a user-declared constructor that is defaulted on its first declaration may be trivial:
struct Foo
{
Foo() = default;
Foo(int, int);
char x;
};
#include <type_traits>
static_assert(std::is_trivially_constructible<Foo>::value, "Works");
The example demonstrates how to define a POD class even in the presence of user-defined (non-default) constructors.
From the standard (12.1), "a default constructor is trivial if it is not user-provided" (plus conditions), and (8.4.2):
A function is user-provided if it is user-declared and not explicitly defaulted or
deleted on its first declaration.
However, note that triviality of a default constructor depends on more than just its declaration and definition. To expand the quote from 12.1:
A default constructor is trivial if it is not user-provided and if:
— its class has no virtual functions (10.3) and no virtual base classes (10.1), and
— no non-static data member of its class has a brace-or-equal-initializer, and
— all the direct base classes of its class have trivial default constructors, and
— for all the non-static data members of its class that are of class type (or array thereof), each such class has a trivial default constructor.