The government's index of leading indicators, designed to forecast turning points in the economy, climbed 1.9 percent in August -- suggesting the nation may be in a recovery -- the Commerce Department said yesterday.

The rise, third in a row, followed increases of 3.7 percent in July and 1 percent in June. The index began climbing in June after declining steadily for a year -- a turnaround that was interpreted as signaling an end to the 1980 recession.

Meanwhile, in an ominous sign, the Agriculture Department reported that farm prices leaped another 1.9 percent in September -- their fifth monthly surge in a row -- and are likely to climb rapidly for months.

Although the jump for September was not nearly as steep as the increases of 3.6 percent for the two previous months, it nevertheless was considered likely to keep supermarket prices rising sharply.

Analysts attributed the increase primarily to higher prices for soybeans, corn, milk, oranges, lettuce and egges, along with tighter meat supplies and a run-up in grain prices because of the summer drought.

The surge in raw agricultural prices is posing a new threat to the government's anti-inflation efforts. Soaring farm prices already have sent producer prices climbing rapidly. And retail prices are not far behind.

The rise in the leading indicators was encouraging to analysts. William A. Cox, the Commerce Department's deputy chief economist, said the report teneded to "lend credence to the assessment that a recovery is occurring."

However, Cox reiterated the department's standard warning that the size of the month-to-month movements in the index does not necessarily bear any relationship to how vigorous an ensuing recovery or slide may be.

Cox said as of now the administration still expects the coming upturn to be "moderate." However, he cautioned, "It would be a mistake to judge the strength of the recovery on the basis of yesterday's figures."

The figures came as, separetely, the Labor Department announced that despite the pickup in economic activity in August there was an increase in layoffs in the nation's manufacturing industries.

The department calculated the layoff rate at 2 percent in August -- up slightly from the 1.7 percent recorded in July but still well below the 2.9 percent rate for June and 3.5 percent for May.

There was no immediate explanation for the increase, which flies in the face of related statistics such as the number of new claims for state unemployment insurance, which has been declining in recent weeks.

The layoff rate, which is among 10 separate statistics used in compiling yesterday's Commerce Department index, was one of three of those components which declined over the month. Cox called it "a curiosity."

Along with the layoff rate, the other negative factors in the leading indicators index were spending for new plant and equipment and new factory orders for durable goods. However, in both cases the declines were small.

The other seven components in the index all posted gains. They include length of work week, vendor performance, change in liquid assets, raw materials prices, stock prices, the money supply and bulding permits.

The August increase pushed the overall leading indicators index to 131 percent of its 1967 average compared with 128.6 percent posted in July.