Is there anyway to use less if statements in my function?
You can use the ternary operation to reduce the vertical space, i.e. x = a if condition else b. This sets x to a if condition is true and to b otherwise.
You can also use a dictionary to map a string to some object, like
the_map = {'a': 4, 'b': 7, 'c': 43}`
x = the_map['c']

This will set x to 43.
Then you can evaluate everything in the initializer instead. So it's not less if checks per se, but it's a bit cleaner, and you can call generate multiple times without needing to do the checks.
import random
import string

class Password:
    def __init__(self, length, string_method, numbers=True, special_chars=False):
        self.length = length
        self.string_method = {
            'upper': string.ascii_uppercase,
            'lower': string.ascii_lowercase,
            'both':  string.ascii_letters
        }[string_method]
        self.numbers = string.digits if numbers else ''
        self.special_chars = string.punctuation if special_chars else ''

    def generate(self, iterations):
        characters = self.string_method + self.numbers + self.special_chars

        # Generating the password
        for p in range(iterations):
            output_password = ''
            for c in range(self.length):
                output_password += random.choice(characters)
            print(output_password)

# Test

password = Password(20, 'lower', True, False)
password.generate(3) # generate the random password 3 times

If you actually want less checks, then just pass in the characters directly. Passing in the length and which characters you want to use into password generator seems perfectly legitimate and straight-forward. This also allows you to do more complex things like generating a password with just vowels, without even numbers, or maybe if you want to exclude certain punctuation.
import random
import string

class Password:
    def __init__(self, length, characters):
        self.length = length
        self.characters = characters

    def generate(self, iterations):
        # Generating the password    
        for p in range(iterations):
            output_password = ''
            for c in range(self.length):
                output_password += random.choice(self.characters)
            print(output_password)

# Test

password = Password(20, string.ascii_lowercase + string.digits)
password.generate(3) # generate the random password 3 times