Zone of Truth is not what you think it is
Zone of Truth only states:
On a failed save, a creature can't speak a deliberate lie while in the radius...
An affected creature is aware of the spell and can thus avoid answering questions to which it would normally respond with a lie. Such a creature can be evasive in its answers as long as it remains within the boundaries of the truth.
This gives whomever is under the effects quite a bit of leeway in answering.
You don't have to divulge the truth, you just can't lie.
But you can twist your words any way you'd like to not be a lie...but to not tell them the truth they are looking.
It's possible a "no" response still isn't lying.
It very much depends on what the 'untruth' is and what it means to you as a character. That is going to entirely be up to you and your DM and it can be made as such that answer "no, i'm not a traitor" is entirely reasonable. Maybe you're not a traitor to your nation, or your family, or to your friends because by being a traitor you're actually helping them. All of this is up for discussion when figuring out what 'being a traitor' actually means.
It's also important to note that whomever is asking is not above suspicion. Throwing that back is always a technique, and a very reasonable one. An innocent person is just as likely to use this tactic.
In order to do this effectively, you will have to work very closely with the DM in terms of your backstory and in terms of what information is given to the other party members that they can act on/ask about.
If they already know the right questions, then it's Game Over, Man.
Yes, they could corner you, but if they've cornered you then they already know how and why you're a traitor. If they're already that knowledgeable about the traitor in their midst, then there is little you can do.
2nd level characters don't have access to 2nd level spell slots.
The spell is also a level 2 spell and would not be accessible by a player character until level 3. No level 2 characters can do this without access to a spell scroll (and they'd risk failure with a DC 12 spellcasting ability check.)
The DM can help
If this is the way the DM wants to set up their story, they can help by providing the uncommon magic item, Ring of Mind Shielding. This item does the following (emphasis mine):
While wearing this ring, you are immune to magic that allows other creatures to read your thoughts, determine whether you are lying...
You can use an action to cause the ring to become invisible until you use another action to make it visible, until you remove the ring, or until you die.
The ring works to prevent magic, like Zone of Truth to do it's job (determine if you are lying) and thus can completely prevent identification of the lie and it can be invisible. Unless they've got See Invisibility active or some way to see it (and to know to look for it), then it's an excellent way to bypass this concern. (Thanks Rubiksmoose!)
Sadly, the language isn't watertight and may still be up to the DM if it will work or not in that way. But if the DM decides it does, and that they give it to the player expressly for this purpose, then it should be fine. If a DM does not think it will, then this is not a viable option.
Player-Player antagonism in a campaign
While not directly related to this, it is most important that your table is open to this type of campaign. If players aren't expecting to be antagonist to each other or actively aren't interested in a table environment of distrust amongst each other, then this may not be a great course of action. Making sure that the table is open to and wanting an intrigue-heavy campaign that includes players possibly playing against other players is something that should be decided at the onset of the campaign prior to start.
You also need to have some sort of (very very flexible) plan for what happens when the traitor (intentionally or otherwise) reveals themselves and now finds themselves in direct and open conflict with the party. Are they ok with their PC dying? Becoming an NPC? etc.