Will this reaction form magnesium carbonate?

The reaction of $\ce{Na2CO3}$ with $\ce{Mg(OH)2}$ will not take place to any significant degree due to solubility issues.

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The problem is that $\ce{Mg(OH)2}$ is about 20 times less soluble than $\ce{MgCO3}$, the latter of which is still only about $\pu{140 mg/L}$ at room temperature. So, even if you were very patient, only a tiny fraction of the $\ce{Mg(OH)2}$ would dissolve then precipitate out as $\ce{MgCO3}$.

The "good" news is that the $\ce{MgCO3}$ synthesis problem doesn't really matter, because it is basically impossible to convert magnesium carbonate (or hydroxide, or oxide) to magnesium metal, outside of a properly equipped laboratory anyway! That is, assuming your home lab doesn't include a $\pu{^{o}C}$ furnace. If you had all the appropriate equipment, all you would need is some carbon and you could do the following (from this Wikipedia article):

$$\ce{MgO(s) + C(s) -> Mg(g) + CO(g)}$$

That is correct, Mg would actually be formed as a gas at $\pu{^{o}C}$.

Sorry to have such negative news for your idea. Please don't hesitate to ask for any clarification in the comments below.

If such a reaction takes place at all it is not very efficient because magnesium hydroxide is not very soluble.

And if you want to get the metallic magnesium neither the carbonate nor the hydroxide will really work. Both decompose at a relatively low temperature to the oxide, which is then solid up to $\pu{\!^\circ C}$. Convert the magnesium to the chloride and melt that, preferably as a eutectic with alkali metal chlorides. Then electrolyze the molten salt. An aqueous solution of magnesium chloride (or anything else containing magnesium) does not work.

You can easily make magnesium carbonate from magnesium oxide by bubbling it in $\ce{CO2}$ gas.

$$\ce{MgO + CO2 &#; MgCO3}$$

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But the problem is that it is impossible to make pure magnesium from magnesium carbonate(@airhuff). Magnesium carbonate decomposes on heating to form magnesium oxide which is very stable.(@Oscar). Instead, you can reduce magnesium oxide into magnesium using reducing agent such as carbon(@airhuff), beryllium or calcium.

$$\ce{MgO + C -> Mg + CO ; ΔH^{0}_{298}= 491.3 kJ / mol}$$

Reaction between magnesium oxide and carbon to form magnesium and carbon monoxide. The reaction proceeds at a temperature above ° C.(Source)

$$\ce{MgO + Be ->[\Delta] BeO + Mg}$$

Reaction interaction beryllium and magnesium oxide to form magnesium and beryllium oxide. The reaction proceeds at a temperature of about ° C.(Source)

$$\ce{Ca + MgO ->[\Delta] CaO + Mg}$$

Reaction between magnesium oxide and calcium to form calcium and magnesium oxide. The reaction proceeds at a temperature of about ° C.(Source)

Carbon is best for this purpose although it takes much more heat than calcium and beryllium. Beryllium takes the lowest amount of heat but I would recommend against it as beryllium is poisonous and expensive. For this reason, calcium can also be used as an alternative.

Also for these reactions, you need a expensive furnace that conducts such type of reaction and can be found only in metallurgical industry. For this reason, these reaction are not part of simple home experiments and cannot be conducted at home. (@airhuff).

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