From the ArrayList.contains(Object o) javadoc:
Returns true if this list contains the specified element. More
  formally, returns true if and only if this list contains at least one
  element e such that (o==null ? e==null : o.equals(e)).
It means, to make your method work correctly, you have to implement the hashcode/equals method in the Account class.
I have created a mock implementation of your code:
No hashcode/equals method defined -> Object class's default hashcode/equals method used which treats each new Object differently. 
import java.util.ArrayList;
import java.util.List;
public class Test{
    static class Account {
        private String AccountName;
        private int AccountNumber;
        private double AccountBalance;
        public Account(String accountName, int accountNumber, double accountBalance)
        {
            this.AccountName = accountName;
            this.AccountNumber = accountNumber;
            this.AccountBalance = accountBalance;
        }
        public Account()
        {
        }
        public String getAccountName() {
            return AccountName;
        }
        public void setAccountName(String accountName) {
            AccountName = accountName;
        }
        public int getAccountNumber() {
            return AccountNumber;
        }
        public void setAccountNumber(int accountNumber) {
            AccountNumber = accountNumber;
        }
        public double getAccountBalance() {
            return AccountBalance;
        }
        public void setAccountBalance(double accountBalance) {
            AccountBalance = accountBalance;
        }       
    }
    static class Bank {
    private List<Account> accountList = new ArrayList<>();
    private String inputStr = "";
    private int inputInt = 0;
    public Bank() {
    }
    public Bank(String string, int i) {
        inputStr = string;
        inputInt = i;
    }
    public boolean addAccount(Account account) {
        if(!accountList.contains(account)) {
            accountList.add(account);
            return true;
        }
        return false;
    }
}
        public static void main(String[] args)
        {
            Bank bank = new Bank("qLOwyONvKsM58ZdV& &yo", 1);
            Account account = new Account("aucchQitgyzLV", 6329668, 479389.0);
            Account account5 = new Account("aucchQitgyzLV", 6329668, 479389.0);
            Account account2 = new Account();
            Account account3 = new Account();
            Account account4 = new Account("sgdgrt", 0, 0.1);
            Account account6 = new Account("sgdgrt", 0, 0.1);
            System.out.println( "account added: "+bank.addAccount(account));
            System.out.println("account5 added: "+bank.addAccount(account5));
            System.out.println( "account2 added: "+bank.addAccount(account2));
            System.out.println("account3 added: "+bank.addAccount(account3));
            System.out.println("account4 added: "+bank.addAccount(account4));
            System.out.println("account6 added: "+bank.addAccount(account6));
        }
}
Result:
  account added: true
  account5 added: true
  account2 added: true
  account3 added: true
  account4 added: true
  account6 added: true
Which obviously not correct.  
If you would insert the following hashcode/equals methods into the Account class: 
@Override
public int hashCode() {
    final int prime = 31;
    int result = 1;
    long temp;
    temp = Double.doubleToLongBits(AccountBalance);
    result = prime * result + (int) (temp ^ (temp >>> 32));
    result = prime * result + ((AccountName == null) ? 0 : AccountName.hashCode());
    result = prime * result + AccountNumber;
    return result;
}
@Override
public boolean equals(Object obj) {
    if (this == obj)
        return true;
    if (obj == null)
        return false;
    if (getClass() != obj.getClass())
        return false;
    Account other = (Account) obj;
    if (Double.doubleToLongBits(AccountBalance) != Double.doubleToLongBits(other.AccountBalance))
        return false;
    if (AccountName == null) {
        if (other.AccountName != null)
            return false;
    } else if (!AccountName.equals(other.AccountName))
        return false;
    if (AccountNumber != other.AccountNumber)
        return false;
    return true;
}  
Result would be:
  account added: true
  account5 added: false
  account2 added: true
  account3 added: false
  account4 added: true
  account6 added: false
So your method would work correctly.