To solve this problem, we need to determine how many quarters have passed between June 12, 2014 and June 12, 2020. Then, we need to calculate how much the initial investment of PHP 240,000 will be worth after that many quarters at an interest rate of 13% compounded quarterly.

To determine the number of quarters that have passed, we first need to determine the number of years that have passed. There are 12 months in a year, so to determine the number of years that have passed, we can divide the number of months that have passed by 12. There are 6 years between 2014 and 2020, so that means there are 6 * 12 = <<6*12=72>>72 months that have passed. Since a quarter consists of 3 months, there have been 72 / 3 = <<72/3=24>>24 quarters that have passed.

To calculate the future value of the initial investment, we can use the formula for compound interest: FV = PV * (1 + r/n) ^ nt, where FV is the future value, PV is the present value (initial investment), r is the interest rate, n is the number of times per year that the interest is compounded, and t is the number of years. In this case, the present value is PHP 240,000, the interest rate is 13% (0.13), the number of times per year that the interest is compounded is 4 (quarterly), and the number of years is 6. Plugging these values into the formula, we get FV = 240,000 * (1 + 0.13/4) ^ (4 * 6) = PHP 613,261.93.

Therefore, the couple will have PHP 613,261.93 on June 12, 2020.