Questar Corp. reported second- quarter net income of $5.75 million, or 28 cents per share, compared with $4.74 million, or 23 cents per share, for the same period last year.

The company said improved earnings by its interstate natural gas transmission operations offset continued weakness in oil and gas exploration and production.Questar's net income for the first six months of 1988 was $32.68 million, or $1.66 per share, compared with $24.30 million, or $1.33 per share, for the 1987 period. The company said the improvement resulted from higher income from natural- gas distribution operations, a lower federal income tax rate and lack of a write-down in the oil and gas exploration area.

R.D. Cash, Questar chairman and chief executive officer, said he is generally pleased with the corporation's earnings performance and underlying strength. "The return to more-normal weather in the first half of the year, combined with higher industrial gas deliveries, has produced the results we anticipated.

"However, oil and gas prices are below expectations," said Cash. "Unless oil or gas prices improve from current levels, or we make additions to our reserves, we face the possibility of another write-down of oil and gas properties under the full-cost accounting rules.

"I expect we have seen the bottom of what is turning out to be the annual summer decline in oil prices and that gas prices will begin to move upward as we approach the winter heating months," he added.

Celsius Energy Co., a Questar exploration and production affiliate, wrote down oil and gas properties in the first quarter of 1987 by $9 million, or 48 cents per share, after taxes, under full-cost accounting rules. Without the write-down, Questar's net income for the first half of 1987 would have been $33,303,000, or $1.81 per share.

Questar's six-month 1988 earnings per share were reduced 9 percent, or 14 cents per share, by the dilutive effect of a 1.7 million share stock sale in June 1987. For the first half of 1988, Questar had an average 19.3 million shares of common stock outstanding compared with 17.7 million for the first half of 1987.

Questar's natural gas distribution operations, conducted by Mountain Fuel Supply Co. had net income of $1.25 million during second quarter 1988 compared with $1.37 million a year earlier.

Mountain Fuel's net income for the first six months of 1988 was $16.84 million, $1.78 million higher than the $15.06 million for the comparable 1987 period.