Under accrual accounting, your income statement should report the amount for the utilities used during the period of the income statement. The utility bills tell you the cost of the gas and electricity used between the meter reading dates appearing on the utility bill. For example, if the meters are read on the last day of the month then the utility bills received in early July will tell you the cost of the utilities used during the month of June. It is this cost that your company should report as expense on its June income statement. The amount of the utility bills with meter reading dates of June 30 should be reported as a current liability (accounts payable) as of June 30.
When the utility bills are based on monthly meter readings other than on the last day of the month, it gets a bit more complicated. For example, if the meters are read on the 20th day of each month, then the utility bill received at the end of June will show the cost of the utilities used between May 21 and June 20. The amount of this bill is also the liability amount as of June 20th. Since the balance sheet is dated June 30th (and the income statement is for the entire month of June), an adjusting entry should be made to record the additional liability for the utilities used from June 20 through June 30.
To have the correct amount on the income statement, you need to report the expense for the 30 or 31 days during the calendar month that the utilities were used. To have the correct liability amount reported on the balance sheet, you need to report the cost of the utilities used through the last day of the month that have not yet been paid. The reason for this complexity is that the utility provides the electricity and gas each day, but it reads the meters only on one day per month. As a result you will always owe the utility some amount, even if you have paid the most recent utility bill. In short you will need to make an adjusting entry to get the financial statements accurate under accrual accounting.
You raised an important point. You should discuss this further with your accountant.