You cannot gain proficiency in this harpoon as it is not a weapon. Ask your GM about training or something else during downtime
There are not stats for the harpoon in question, all we have is the stats for the device that fires it, the harpoons are merely ammunition and not themselves weapons.
So there would be no way to gain proficiency in them, except as improvised weapons which is not particularly helpful. Even if we assumed they were weapons any method of gaining proficiency in weapons won't apply to harpoons.
For example: the Weapon Master feat states:
You have practiced extensively with a variety of weapons, gaining the following benefits:
- Increase your Strength or Dexterity by 1, to a maximum of 20.
- You gain proficiency with four weapons of your choice. Each one must be a simple or a martial weapon.
Note that this is restricted to simple and martial weapons, and a harpoon is neither of these as we do not have stats for it.
There are other class/subclass specific methods such as a Pact of the Blade Warlock who can use their Pact Boon feature which states:
You can use your action to create a pact weapon in your empty hand. You can choose the form that this melee weapon takes each time you create it (see the Weapons section for weapon options). You are proficient with it while you wield it.
But again, this requires you to make a weapon from the Weapons section, and the harpoon does not exist there.
I would recommend using the Downtime rules and discussing with your GM. These rules state (PHB page 187):
Downtime activities other than the ones presented below are possible. If you want your character to spend his or her downtime performing an activity not covered here, discuss it with your DM [...]
You can spend time between adventures learning a new language or training with a set of tools. Your DM might allow additional training options.
First, you must find an instructor willing to teach you. The DM determines how long it takes, and whether one or more ability checks are required.
The training lasts for 250 days and costs 1 gp per day. After you spend the requisite amount of time and money, you learn the new language or gain proficiency with the new tool.
The DMG (page 231) also states regarding Downtime:
A character who agrees to training as a reward must spend downtime with the trainer (see chapter 6 for more information on downtime activities). In exchange, the character is guaranteed to receive a special benefit. Possible training benefits include the following:
- The character gains inspiration daily at dawn for 1d4 + 6 days.
- The character gains proficiency in a skill.
- The character gains a feat.