It depends. Since he was "placed" with you on 7/6/2020 he would not meet the rule number 3 (below) for 2020 as a qualifying child. To be a foster child he must have been placed by an authorized placement agency or by judgment, decree, or other order of any court of competent jurisdiction.
However, if he was placed with you by an authorized placement agency or by judgment, decree, or other order of any court of competent jurisdiction, then he could meet the qualifications for an "other dependent" (see below for other qualifying dependent) for 2020.
You should add the dependent to your return and answer the questions in TurboTax and the program will let you know if you can claim or not.
For 2021 again if he was placed with you by an authorized placement agency or by judgment, decree, or other order of any court of competent jurisdiction and stays with you for 6 months and 1 day in 2021, then he would be a qualifying child. If less than 1/2 the year then he would still be an "Other qualifying dependent".
If he was not placed by an authorized placement agency, then he needs to live with you all year to qualify as an "other dependent".
Qualifying child:
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The child must be your son, daughter, stepchild, foster child, brother, sister, half brother, half sister, stepbrother, stepsister, or a descendant of any of them.
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The child must be (a) under age 19 at the end of the year and younger than you (or your spouse if filing jointly), (b) under age 24 at the end of the year, a student, and younger than you (or your spouse if filing jointly), or (c) any age if permanently and totally disabled.
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The child must have lived with you for more than half of the year.
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The child must not have provided more than half of his or her own support for the year.
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The child must not be filing a joint return for the year (unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid).
Other Qualifying dependent:
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The person can't be your qualifying child or the qualifying child of any other taxpayer.
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The person either (a) must be related to you in one of the ways listed under Relatives who don't have to live with you , or (b) must live with you all year as a member of your household (and your relationship must not violate local law).
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The person's gross income for the year must be less than $4,300.
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You must provide more than half of the person's total support for the year.
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