hasi no mukou ni sizukana kouen ga arimasu.

橋の向こう means the other side of the bridge or beyond the bridge.

• 橋 = bridge
• の links nouns
• 向こう = the other side / the far side / beyond

So A の 向こう means the other side of A.

Examples:

• 川の向こう = the other side of the river
• 山の向こう = beyond the mountain

Here, の connects two nouns.

It is not always possession. It often works like of, at, or just a linking marker in English.

So:

• 橋の向こう = the bridge’s other side → more naturally, the other side of the bridge

This is a very common pattern:

• 駅の前 = in front of the station
• 家の中 = inside the house
• 学校の近く = near the school

に is used because this sentence is about existence in a location.

With あります and います, the place where something exists is usually marked by に:

• 場所 に もの が あります
• 場所 に 人 が います

So:

• 橋の向こうに = on the other side of the bridge

Do not use で here.
で marks the place where an action happens, but あります is not an action like eat, study, or play. It expresses existence.

Compare:

• 公園にベンチがあります。 = There is a bench in the park.
• 公園で遊びます。 = I play in the park.

Because 静か is a na-adjective.

There are two main adjective types in Japanese:

• い-adjectives: like 高い, 新しい
• な-adjectives: like 静か, きれい, 便利

When a na-adjective comes before a noun, you add な:

• 静かな公園 = a quiet park

But when it appears at the end of a sentence, you do not use な:

• 公園は静かです。 = The park is quiet.

So:

• before noun: 静かな公園
• sentence end: 公園は静かです

が is commonly used in existence sentences with あります / います to mark the thing that exists.

Pattern:

• 場所 に もの が あります
• 場所 に 人 が います

So here:

• 公園があります = there is a park

Using が makes sense because the park is the thing being identified as existing.

If you used は, the nuance would change. For example:

• 公園はあります。 can sound like As for a park, there is one or contrastive, depending on context.

In simple existence statements, が is the most natural choice.

Because 公園 is not a living thing.

Japanese usually uses:

• あります for inanimate things
• います for animate beings, especially people and animals

So:

• 公園があります。 = There is a park.
• 犬がいます。 = There is a dog.

Even though a park is a place, grammatically it is treated as a thing that exists, so あります is correct.

Japanese does not need a topic in every sentence.

This sentence is a straightforward existence statement:

• 橋の向こうに静かな公園があります。
• There is a quiet park on the other side of the bridge.

It sounds natural without は because the main purpose is to introduce or point out the existence of the park.

You could add a topic in a larger context, for example:

• 橋の向こうには静かな公園があります。

This adds emphasis or contrast, something like:

• On the other side of the bridge, there is a quiet park.

So the version without は is simple and neutral.

Japanese word order is more flexible than English, but some patterns are more natural than others.

This sentence follows a very common order:

• location + に
• noun + が
• verb

So:

• 橋の向こうに 静かな公園が あります

That is very natural.

You may also see slight variations for emphasis, but the verb usually comes at the end:

• 静かな公園が橋の向こうにあります。

This is still grammatical, but the original version sounds more like normal description of location.

Not always.

In this sentence, it is physical:

• 橋の向こう = the other side of the bridge

But 向こう can also mean:

• over there
• the other side
• beyond
• the other party / the other person / the other group, depending on context

Examples:

• 向こうに山が見える。 = You can see mountains over there.
• 向こうのチーム = the other team

So the exact nuance depends on context, but in your sentence it is clearly a location beyond the bridge.

Japanese does not have articles like a, an, and the.

So 静かな公園 can mean:

• a quiet park
• the quiet park

The exact meaning depends on context.

In this sentence, English naturally uses a quiet park because the park is being introduced for the first time:

• There is a quiet park on the other side of the bridge.

Japanese leaves that unstated.

Yes, that is a very natural translation.

Depending on context, these can all work:

• There is a quiet park on the other side of the bridge.
• There is a quiet park across the bridge.
• Beyond the bridge, there is a quiet park.

The Japanese does not force only one English phrasing. The core idea is that the park is located past the bridge / on the far side of the bridge.