What I'm wanting to prove is what it says in the title of the question.
Or more formally:
$P=(V_p,E_p)$, $Q=(V_q,E_q)$ are two trails of the max length possible / G=(V,E) a connected graph, $(V_p \cup V_q) \subseteq V,(E_p \cup E_q) \subseteq E \Rightarrow$ $\exists v_0 \in V / v_0 \in (V_p \cap V_q)$
My idea was: Suppose you create two trails (P and Q) and they are of the max length possible (they are both on the same connected graph) and they are also disconnected between them. That can't be never true (the fact they are of the max length possible) because if you say it is a connected graph then you can always find a path between any edge of it(that's the definition of connected graph).
And knowing that, you can't never say that you can obtain two trails of it's max length possible and disconnected because you can continue drawing the trail, join them at some edge and doing that you obtain a longer trail.
Sorry if I expressed my idea a bit confusing but I want to let you know that even in Spanish (my native language) is hard for me to write a demonstration right now (I've just started doing these kind of mathematical problems: proving things, this year) so in English is even harder for me.
I would like to know if my reasoning is right or I'm failing at something or maybe if there is a better way to prove this. Or if I should reorder the way I'm proving this...
What I mean by TRAILS is a walk that can repeat vertices but not edges and it's open (in spanish called recorrido), definition taken from the book Ralph Grimaldi - Discrete and Combinatorial Mathematics - 5ed :