No, you cannot both claim the same child on your separate returns. Only one of you claims him.
Why are you filing separate returns, though?
Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will each receive the $4000 personal exemption, plus the married filing jointly standard deduction of $12,600 (add $1250 for each spouse over the age of 65). You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit.
If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. In many cases you will not be able to take the child and dependent care credit. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. If you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice