Yes. Instead of working in $C$, you can work in presheaves $[C^{op}, \text{Set}]$ on $C$ using the Yoneda embedding. There is always a terminal presheaf given by sending every object $c \in C$ to $1 \in \text{Set}$ (whether or not it's representable by an object in $C$), and so you can make the following definitions using it. 

> **Definition:** A global element of $c \in C$ is a natural transformation $1 \to \text{Hom}(-, c)$ of presheaves.
>
> **Definition:** A constant morphism $f : c \to d$ is a morphism such that the induced morphism $\text{Hom}(- , f) : \text{Hom}(-, c) \to \text{Hom}(-, d)$ factors through the terminal presheaf.

Because the Yoneda embedding is fully faithful and preserves all limits that exist in $C$, these definitions are guaranteed to reproduce the usual definitions if $C$ does in fact have a terminal object. 

Unwinding these definitions, we get the following.

> **Definition:** A global element of $c$ is a choice, for each object $c' \in C$, of a morphism $f_{c'} : c' \to c$ such that, for every morphism $g : c' \to c''$, we have $f_{c''} g = f_{c'}$.
>
> **Definition (edited, 7/6/16):** A constant morphism $f : c \to d$ is a morphism such that there is a choice $f_{d'} : d' \to d$ of global element of $d$ in the above sense such that, for every morphism $g : c' \to c$, we have $f g = f_{c'}$.